For high rate systems, the hyperspherical and hyperdiamond shaping reduce the unclipped peak power by 5 to 7 dB with no loss of data rate and with a symbol error rate that is virtually identical to that of conventional systems. With a clipping rate of 10-8, they still provide up to 2.8 dB of peak-power reduction. Algebraic shaping reduces the unclipped peak power by 8 to 10 dB and about 2.9 dB at a clipping rate of 10-8 for 16-channel systems. Moreover, algebraic shaping asymptotically provides an unlimited amount of peak-power reduction as the number of channels and the constellation size increase. It also approaches the PAR characteristics of a single-carrier system (such as QAM or PAM) as the number of constellation points per channel increases. In addition, we present an O( N log2N) algorithm using a discrete Hadamard transform-based OFDM and introduce a method for incorporating arbitrary lattice codes inside the shaped boundary.